# Is a paradox a counterexample to $p \lor \neg p$ always being true?

On Silicy everyone lies all the time, and if you are not from Silicy, you speak the truth all the times. Call the proposition "I am from Silicy" $p$.

Then $p$ is not true, because then he is from Silicy, thus lies.

But $p$ cannot be false either because then he is not from Silicy, which means he always speaks true, which contradicts that $p$ is false.

Can someone explain to me where I go wrong here? My knowledge of logic is very limited: I know truth tables, the logical symbols and some tautologies such as de Morgan's laws, and that is pretty much it.

• Such self-referential phrases ($p$ is about $p$) do not fit in the framework of Boolean algebra, they are not acceptable propositions. – Yves Daoust Aug 2 '17 at 10:39
• @YvesDaoust I think your comment should be an answer – TZakrevskiy Aug 2 '17 at 10:47
• As @YvesDaoust points out, you cannot assume something that contradicts with your assumptions, so statement "On silicy everyone lies" is not a valid nor acceptable statement.I mean of course you can assume such thing, but if you do that, you can derive almost anything from it. – onurcanbektas Aug 2 '17 at 10:49
• How did Crete become Sicily here? (I suspect Vizzini may be to blame). – Henning Makholm Aug 2 '17 at 11:37
• @HenningMakholm: in Lieland, Sicily is called Silicy. – Yves Daoust Aug 2 '17 at 12:15

Such self-referential phrases ($p$ is about $p$) do not fit in the framework of Boolean algebra/propositional logic, they are not acceptable propositions.
A proposition must have a well-defined truth value, either $\text{true}$ or $\text{false}$, and $\text{true}\lor\lnot\text{true}$, $\text{false}\lor\lnot\text{false}$ do hold.
Proposition $p$: Someone from Sicily says "I lie". Your first assertion is correct, this cannot hold if we assume that everyone on Sicily lies.
Therefore, $p$ is false. Someone from Sicily, in this case, did not say "I lie".